Why are there gaps in between my model?
You missed at least two things while following the tutorial.

At 11:15 Steve explains that you have to set the Array Modifier to Fit Curve. Your screenshot shows you've set it to Fit Length.
The second thing is, after adding the Subdivision Surface Modifier he explains why it doesn't look smooth enough yet and at 12:25 he enables the option Merge in the Array Modifier.

Maybe these two things might help to make it look right.